Acoustic traveling waves in thermoviscous perfect gases: Kinks, acceleration waves, and shocks under the Taylor–Lighthill balance

Abstract

We study one-dimensional traveling wave phenomena in thermoviscous perfect gases with constant material properties. First, we summarize the known forms of the solution of a classic acoustic signaling problem based on the linearized theory. Next, we review several weakly-nonlinear models, all of which admit traveling wave solutions (TWS)s in the form of classical Taylor shocks, and note some of their features. We then consider traveling waves under the fully-nonlinear theory of gas dynamics, and derive a new third-order equation of motion valid for arbitrary Mach number values. Focusing on the special cases of (a) inviscid, thermally-conducting, and (b) viscous, nonthermally-conducting (i.e., strictly adiabatic) perfect gases, exact and/or numerical solutions of this nonlinear ordinary differential equation (ODE) are determined, asymptotic expressions presented, and critical values of the physical parameters identified. It is shown that, in addition to kinks, the fully-nonlinear theory allows for discontinuous solutions (i.e., shocks) and solutions that exhibit acceleration waves.

Acceleration waves in a nonlinear Biot theory of porous media

“…There is much interest in the propagation of waves in porous and acoustic media, see e.g. Biot [1], Brunnhuber and Jordan [2], Christov [3], Christov and Jordan [4], Christov et al [5], Ciarletta and Straughan [6][7][8], Jordan [9][10][11][12][13][14], Jordan and Puri [15], Jordan and Saccomandi [16], Jordan et al [17,18], Paoletti [19], Rossmanith and Puri [20,21], Wei and Jordan [22]. This interest is driven by the many real life applications this topic has.…”

Nonlinear acoustics and shock formation in lossless barotropic Green--Naghdi fluids

“…Although we consider only the lossless case in this work, viscous and thermoviscous acoustic wave propagation under the classical continuum theories (e.g., Navier-Stokes-Fourier) [54,55,56,69,36,15,64,41,23,33,34,19,65], as well as in inviscid but thermally relaxing gases [37], has also received significant attention in the literature.…”

Thermal effects on nonlinear acceleration waves in the Biot theory of porous media

“…The topic of wave propagation in porous and acoustic media is one of great interest in the current research literature, see e.g. Biot [1], Brunnhuber and Jordan [2], Christov [3], Christov and Jordan [4], Christov et al [5], Ciarletta and Straughan [6][7][8], Jordan [9][10][11][12][13][14], Jordan and Puri [15], Jordan and Saccomandi [16], Jordan et al [17,18], Paoletti [19], Rossmanith and Puri [20,21], Wei and Jordan [22].…”